If the angle of elevation of the sun changed from `45^(@)` to `60^(@)` , then the length of the shadow of a pillar decreases by 10 m .The height of the pillar is :

Options

A. `5(3-sqrt(3))` metre
B. `5(sqrt(3)+1)` metre
C. `15(sqrt(3)+1)` metre
D. `5(3+sqrt(3))` metre

Answer

To solve the problem, we need to determine the height of the pillar based on the change in the angle of elevation of the sun and the change in the length of the shadow. Let's break it down step by step.

### Step 1: Define the Variables
Let:
- \( h \) = height of the pillar
- \( x \) = original length of the shadow when the angle of elevation is \( 45^\circ \)

### Step 2: Use Trigonometric Ratios for the First Angle
When the angle of elevation is \( 45^\circ \):
- The tangent of the angle is given by:
\[ \tan(45^\circ) = 1 \]
- Therefore, we can write:
\[ \frac{h}{x} = 1 \implies h = x \]

### Step 3: Use Trigonometric Ratios for the Second Angle
When the angle of elevation changes to \( 60^\circ \):
- The tangent of the angle is given by:
\[ \tan(60^\circ) = \sqrt{3} \]
- The length of the shadow decreases by 10 m, so the new length of the shadow is \( x - 10 \). We can write:
\[ \frac{h}{x - 10} = \sqrt{3} \implies h = \sqrt{3}(x - 10) \]

### Step 4: Set Up the Equation
Now we have two expressions for \( h \):
1. \( h = x \)
2. \( h = \sqrt{3}(x - 10) \)

Setting these two equations equal to each other:
\[ x = \sqrt{3}(x - 10) \]

### Step 5: Solve for \( x \)
Expanding the equation:
\[ x = \sqrt{3}x - 10\sqrt{3} \]
Rearranging gives:
\[ x - \sqrt{3}x = -10\sqrt{3} \]
Factoring out \( x \):
\[ x(1 - \sqrt{3}) = -10\sqrt{3} \]
Thus,
\[ x = \frac{-10\sqrt{3}}{1 - \sqrt{3}} \]

### Step 6: Rationalize the Denominator
To rationalize the denominator, multiply the numerator and denominator by the conjugate:
\[ x = \frac{-10\sqrt{3}(1 + \sqrt{3})}{(1 - \sqrt{3})(1 + \sqrt{3})} \]
Calculating the denominator:
\[ (1 - \sqrt{3})(1 + \sqrt{3}) = 1 - 3 = -2 \]
Thus:
\[ x = \frac{-10\sqrt{3}(1 + \sqrt{3})}{-2} = \frac{10\sqrt{3}(1 + \sqrt{3})}{2} = 5\sqrt{3}(1 + \sqrt{3}) \]

### Step 7: Find the Height \( h \)
Now substituting \( x \) back into \( h = x \):
\[ h = 5\sqrt{3}(1 + \sqrt{3}) = 5\sqrt{3} + 15 \]

### Final Answer
The height of the pillar is:
\[ h = 5\sqrt{3} + 15 \text{ meters} \]